Autocorrelation polynomial of complex numbers

Question

I have read the article "On the ambiguity of the image reconstruction problem", which is about the phase retrieval problem (but this is not relevant to my question).

Now to my problem: I have |z|=1, where z is a complex number. This implies that $z^{-1}$=z*. I want to build the "autocorrelation polynomial" (here: square of the absolute value): Q(z)=p(z)p(z*)=p(z)p($z^{-1}$). According to the author if p(z)= $\prod_{j=1}^{m}$(z-$z_j$) than Q(z)= $\prod_{j=1}^{m}$(z-$z_j$)(z-$z_j^{-1}$). Can someone tell me, how I get this Q(z). I don't understand why I can take the factor (z-$z_j^{-1}$) because $z_{j}$ have not to be on the unit circle.

Thank you for your help!!

Answer 1

Extremely confusing and definitely wrong as written: If $Q(z) = p(z)p(1/z)$, then $Q$ isn't a polynomial of degree $2m$, but...

By hypothesis, $$p(z) = 0\implies z = z_1,\dots,z_m.$$ Then, $$p(1/z) = 0\implies 1/z = z_1,\dots,z_m\implies z = 1/z_1,\dots,1/z_m.$$